The water cycle, also known as the hydrologic cycle or H2O cycle, describes the continuous movement of water on, above and below the surface of the Earth. Water can change states among liquid, vapor, and ice at various places in the water cycle. Although the balance of water on Earth remains fairly constant over time, individual water molecules can come and go, in and out of the atmosphere. The water moves from one reservoir to another, such as from river to ocean, or from the ocean to the atmosphere, by the physical processes of evaporation, condensation, precipitation, infiltration, runoff, and subsurface flow. In so doing, the water goes through different phases: liquid, solid, and gas.The hydrologic cycle also involves the exchange of heat energy, which leads to temperature changes. For instance, in the process of evaporation, water takes up energy from the surroundings and cools the environment. Conversely, in the process of condensation, water releases energy to its surroundings, warming the environment.And for second one:The carbon cycle is the biogeochemical cycle by which carbon is exchanged among the biosphere, pedosphere, geosphere, hydrosphere, and atmosphere of the Earth. It is one of the most important cycles of the earth and allows for carbon to be recycled and reused throughout the biosphere and all of its organisms.

The cycle of evaporation and condensation that controls the distribution of the earth's water as it evaporates from bodies of water, condenses, precipitates, and returns to those bodies of water. Also called hydrologic cycle.
Cycle that involves the continuous circulation of water in the Earth-atmosphere system. Water is transferred from the oceans through the atmosphere to the continents and back to the oceans by means of evaporation, transpiration, precipitation, interception, infiltration, subterranean percolation, overland flow, runoff, and other complex processes. Although the total amount of water within the cycle remains essentially constant, its distribution among the various processes is continually changing.

Carbon cycle is the biogeochemical cycle by which carbon is exchanged among the biosphere, pedosphere, geosphere, hydrosphere, and atmosphere of the Earth. It is one of the most important cycles of the earth and allows for carbon to be recycled and reused throughout the biosphere and all of its organisms.

I think the best answer to this question is given by Dictionary.com. It is brief and gives an excellent summary of the process in both fields of science.
carbon cycle
–noun
1.
Ecology . the circulation of carbon atoms in the biosphere as a result of photosynthetic conversion of carbon dioxide into complex organic compounds by plants, which are consumed by other organisms: the carbon returns to the atmosphere in the form of carbon dioxide as a result of respiration, decay by fungi, bacteria, etc., and combustion of fossil fuels.
2.
Astrophysics . a cycle of nuclear transformations in the interiors of the stars by means of which hydrogen is gradually converted into helium with the release of nuclear energy.